Three-body bound states in a harmonic waveguide with cylindrical symmetry

Highly-elongated quasi-one-dimensional cold atom samples have been studied extensively over the past years experimentally and theoretically. This work determines the energy spectrum of two identical fermions and a third distinguishable particle as functions of the mass ratio $\kappa$ and the free-space $s$-wave scattering length $a_{3\text{D}}$ between the identical fermions and the distinguishable third particle in a cylindrically symmetric waveguide whose symmetry axis is chosen to be along the $z$-axis. We focus on the regime where the mass of the identical fermions is equal to or larger than that of the third distinguishable particle. Our theoretical framework accounts explicitly for the motion along the transverse confinement direction. In the regime where excitations in the transverse direction are absent (i.e., for states with projection quantum number $M_{\text{rel}}=0$), we determine the binding energies for states with odd parity in $z$. These full three-dimensional energies deviate significantly from those obtained within a strictly one-dimensional framework when the $s$-wave scattering length is of the order of or smaller than the oscillator length in the confinement direction. If transverse excitations are present, we predict the existence of a new class of universal three-body bound states with $|M_{\text{rel}}|=1$ and positive parity in $z$. These bound states arise on the positive $s$-wave scattering length side if the mass ratio $\kappa$ is sufficiently large. Implications of our results for ongoing cold atom experiments are discussed.Comment: 9 figure

Efimov physics and the three-body parameter for shallow van der Waals potentials

Extremely weakly-bound three-boson systems are predicted to exhibit intriguing universal properties such as discrete scale invariance. Motivated by recent experimental studies of the ground and excited helium trimers, this work analyzes the three-body parameter and the structural properties of three helium atoms as the s-wave scattering length is tuned artificially. Connections with theoretical and experimental studies of the Efimov scenario as it pertains to cold atom systems are made.Comment: 10 pages, 5 figures in Few-Body Systems (2015

Hyperspherical explicitly correlated Gaussian approach for few-body systems with finite angular momentum

Within the hyperspherical framework, the solution of the time-independent Schroedinger equation for a n-particle system is divided into two steps, the solution of a Schroedinger like equation in the hyperangular degrees of freedom and the solution of a set of coupled Schroedinger like hyperradial equations. The solutions to the former provide effective potentials and coupling matrix elements that enter into the latter set of equations. This paper develops a theoretical framework to determine the effective potentials, as well as the associated coupling matrix elements, for few-body systems with finite angular momentum L=1 and negative and positive parity. The hyperangular channel functions are expanded in terms of explicitly correlated Gaussian basis functions and relatively compact expressions for the matrix elements are derived. The developed formalism is applicable to any n; however, for n greater or equal to 6, the computational demands are likely beyond present-day computational capabilities. A number of calculations relevant to cold atom physics are presented, demonstrating that the developed approach provides a computationally efficient means to solving four-body bound and scattering problems with finite angular momentum on powerful desktop computers. Details regarding the implementation are discussed

Path integral Monte Carlo determination of the fourth-order virial coefficient for unitary two-component Fermi gas with zero-range interactions

The unitary equal-mass Fermi gas with zero-range interactions constitutes a paradigmatic model system that is relevant to atomic, condensed matter, nuclear, particle, and astro physics. This work determines the fourth-order virial coefficient $b_4$ of such a strongly-interacting Fermi gas using a customized \textit{ab initio} path integral Monte Carlo (PIMC) algorithm. In contrast to earlier theoretical results, which disagreed on the sign and magnitude of $b_4$, our $b_4$ agrees within error bars with the experimentally determined value, thereby resolving an ongoing literature debate. Utilizing a trap regulator, our PIMC approach determines the fourth-order virial coefficient by directly sampling the partition function. An on-the-fly anti-symmetrization avoids the Thomas collapse and, combined with the use of the exact two-body zero-range propagator, establishes an efficient general means to treat small Fermi systems with zero-range interactions.Comment: 5 pages (2 figures) + 5 pages (1 figure). Added a table, included more discussion about the fitted function, corrected a few typos, and updated Fig.

Incorporating exact two-body propagators for zero-range interactions into NN-body Monte Carlo simulations

Ultracold atomic gases are, to a very good approximation, described by pairwise zero-range interactions. This paper demonstrates that $N$-body systems with two-body zero-range interactions can be treated reliably and efficiently by the finite temperature and ground state path integral Monte Carlo approaches, using the exact two-body propagator for zero-range interactions in the pair product approximation. Harmonically trapped one- and three-dimensional systems are considered. A new propagator for the harmonically trapped two-body system with infinitely strong zero-range interaction, which may also have applications in real time evolution schemes, is presented.Comment: 11 pages, 6 figure

Microscopic Superfluidity in Bose Gases: From 3D to 1D

The superfluid fraction of ideal and interacting inhomogeneous Bose gases with varying asymmetry is investigated at finite temperature using well-known properties of the harmonic oscillator as well as the essentially exact microscopic path integral Monte Carlo method. We find that the superfluid fraction (i) is essentially independent of the interaction strength for all temperatures considered, (ii) changes profoundly as the effective dimensionality is varied from three- to one-dimensional, (iii) is approximately equal to the condensate fraction N0/N for spherical Bose gases, and (iv) deviates dramatically from N0/N for highly-elongated Bose gases.Comment: 4 pages, 3 fig

Abnormal Superfluid Fraction of Harmonically Trapped Few-Fermion Systems

Superfluidity is a fascinating phenomenon that, at the macroscopic scale, leads to dissipationless flow and the emergence of vortices. While these macroscopic manifestations of superfluidity are well described by theories that have their origin in Landau's two-fluid model, our microscopic understanding of superfluidity is far from complete. Using analytical and numerical \textit{ab initio} approaches, this paper determines the superfluid fraction and local superfluid density of small harmonically trapped two-component Fermi gases as a function of the interaction strength and temperature. At low temperature, we find that the superfluid fraction is, in certain regions of the parameter space, negative. This counterintuitive finding is traced back to the symmetry of the system's ground state wave function, which gives rise to a diverging quantum moment of inertia $I_{\text{q}}$. Analogous abnormal behavior of $I_{\text{q}}$ has been observed in even-odd nuclei at low temperature. Our predictions can be tested in modern cold atom experiments.Comment: 5 pages, 4 figure

Harmonically trapped Fermi gas: Temperature dependence of the Tan contact

Ultracold atomic gases with short-range interactions are characterized by a number of universal species-independent relations. Many of these relations involve the two-body Tan contact. Employing the canonical ensemble, we determine the Tan contact for small harmonically trapped two-component Fermi gases at unitarity over a wide range of temperatures, including the zero and high temperature regimes. A cluster expansion that describes the properties of the N-particle system in terms of those of smaller subsystems is introduced and shown to provide an accurate description of the contact in the high temperature regime. Finite-range corrections are quantified and the role of the Fermi statistics is elucidated by comparing results for Fermi, Bose and Boltzmann statistics.Comment: 5 figures (several subfigures

Path integral Monte Carlo ground state approach: Formalism, implementation, and applications

Monte Carlo techniques have played an important role in understanding strongly-correlated systems across many areas of physics, covering a wide range of energy and length scales. Among the many Monte Carlo methods applicable to quantum mechanical systems, the path integral Monte Carlo approach with its variants has been employed widely. Since semi-classical or classical approaches will not be discussed in this review, path integral based approaches can for our purposes be divided into two categories: approaches applicable to quantum mechanical systems at zero temperature and approaches applicable to quantum mechanical systems at finite temperature. While these two approaches are related to each other, the underlying formulation and aspects of the algorithm differ. This paper reviews the path integral Monte Carlo ground state (PIGS) approach, which solves the time-independent Schroedinger equation. Specifically, the PIGS approach allows for the determination of expectation values with respect to eigen states of the few- or many-body Schroedinger equation provided the system Hamiltonian is known. The theoretical framework behind the PIGS algorithm, implementation details, and sample applications for sermonic systems are presented.Comment: 56-page tutorial to be published in JPB; a related PIMC code, authored by Yangqian Yan, can be found at https://github.com/yangqian/PIMCcod

Spin structure of harmonically trapped one-dimensional atoms with spin-orbit coupling

We introduce a theoretical approach to determine the spin structure of harmonically trapped atoms with two-body zero-range interactions subject to an equal mixture of Rashba and Dresselhaus spin-orbit coupling created through Raman coupling of atomic hyperfine states. The spin structure of bosonic and fermionic two-particle systems with finite and infinite two-body interaction strength $g$ is calculated. Taking advantage of the fact that the $N$-boson and $N$-fermion systems with infinitely large coupling strength $g$ are analytically solvable for vanishing spin-orbit coupling strength $k_{so}$ and vanishing Raman coupling strength $\Omega$, we develop an effective spin model that is accurate to second-order in $\Omega$ for any $k_{so}$ and infinite $g$. The three- and four-particle systems are considered explicitly. It is shown that the effective spin Hamiltonian, which contains a Heisenberg exchange term and an anisotropic Dzyaloshinskii-Moriya exchange term, describes the transitions that these systems undergo with the change of $k_{so}$ as a competition between independent spin dynamics and nearest-neighbor spin interactions.Comment: 23 pages, 10 figure